Lets say we are interested in some unknown variable x. We poll a x at different intervals and get a set of values that x has been. However we know that if we poll x enough times we will eventually get repetition since x is random. What can be gained from knowing the upper bound of x.
A brief example: Lets say I want to take a snapshot of the magnitude of the acceleration of a car. I can do this at any time. I might take 10, 50 or 1000 snapshots. None of these can defiantly tell me the upper bound of the acceleration, although the more samples I take the more confident I can be that I've either captured it, or am near it. Now, lets say I know the local geography, specifically the steepest hills in the area. I also have the cars schematic. With these two pieces of information I can (in theory) calculate the maximum acceleration of the car. Is this beneficial in any way to my statistics? The lower bound is clearly 0, that is known before any samples or other information. What new data can I learn from knowing the upper bound?
Originally my intuition was that the upper bound would help me gain details of the population size, but they now seem completely unrelated.
More detail on my problem:
Lets assume I want to predict the number of car crashes on a given stretch of road. If I say there's a small chance of any 2 cars colliding, based on the make of the car, the relative speed etc. then I could simulate every possible eventuality. For example, car 'A' collides with one car in a set of simulations, but the car it collides with ranges through the population of cars. There'll also be the case where car 'A' collides with car 'B' and in the same simulation car 'C' collides with car 'D' (ie. 2 collisions in 1 simulation) this may be less likely but still possible. Now given the total population of cars, and the number of time steps in the simulation I could work out how many possible different results there are. Some results will occur more than once in a random collection of simulations, but in terms of unique simulations there should be a fixed, finite number. But if I can calculate this number what, if anything, does it teach me about the problem?